Properties

Label 3400.1.y.a
Level $3400$
Weight $1$
Character orbit 3400.y
Analytic conductor $1.697$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3400,1,Mod(251,3400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3400.251");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3400.y (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.69682104295\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 136)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.314432.1
Artin image: $C_4^2:C_2^2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - i q^{2} + ( - i + 1) q^{3} - q^{4} + ( - i - 1) q^{6} + i q^{8} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} + ( - i + 1) q^{3} - q^{4} + ( - i - 1) q^{6} + i q^{8} - i q^{9} + (i + 1) q^{11} + (i - 1) q^{12} + q^{16} - i q^{17} - q^{18} - i q^{19} + ( - i + 1) q^{22} + (i + 1) q^{24} + q^{27} - i q^{32} + q^{33} - q^{34} + i q^{36} - 2 q^{38} + ( - i - 1) q^{41} + ( - i - 1) q^{44} + ( - i + 1) q^{48} + i q^{49} + ( - i - 1) q^{51} + ( - 2 i - 2) q^{57} - q^{64} + ( - 2 i + 1) q^{66} + i q^{68} + q^{72} + ( - i + 1) q^{73} + 2 i q^{76} + q^{81} + (i - 1) q^{82} + (i - 1) q^{88} + ( - i - 1) q^{96} + (i - 1) q^{97} + q^{98} + ( - i + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{4} - 2 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{11} - 2 q^{12} + 2 q^{16} - 2 q^{18} + 2 q^{22} + 2 q^{24} + 4 q^{33} - 2 q^{34} - 4 q^{38} - 2 q^{41} - 2 q^{44} + 2 q^{48} - 2 q^{51} - 4 q^{57} - 2 q^{64} + 2 q^{72} + 2 q^{73} + 2 q^{81} - 2 q^{82} - 2 q^{88} - 2 q^{96} - 2 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3400\mathbb{Z}\right)^\times\).

\(n\) \(1601\) \(1701\) \(2177\) \(2551\)
\(\chi(n)\) \(-i\) \(-1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.00000i
1.00000i
1.00000i 1.00000 + 1.00000i −1.00000 0 −1.00000 + 1.00000i 0 1.00000i 1.00000i 0
3251.1 1.00000i 1.00000 1.00000i −1.00000 0 −1.00000 1.00000i 0 1.00000i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
17.c even 4 1 inner
136.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3400.1.y.a 2
5.b even 2 1 136.1.j.a 2
5.c odd 4 1 3400.1.bc.a 2
5.c odd 4 1 3400.1.bc.b 2
8.d odd 2 1 CM 3400.1.y.a 2
15.d odd 2 1 1224.1.s.a 2
17.c even 4 1 inner 3400.1.y.a 2
20.d odd 2 1 544.1.n.a 2
40.e odd 2 1 136.1.j.a 2
40.f even 2 1 544.1.n.a 2
40.k even 4 1 3400.1.bc.a 2
40.k even 4 1 3400.1.bc.b 2
85.c even 2 1 2312.1.j.b 2
85.f odd 4 1 3400.1.bc.b 2
85.i odd 4 1 3400.1.bc.a 2
85.j even 4 1 136.1.j.a 2
85.j even 4 1 2312.1.j.b 2
85.m even 8 2 2312.1.e.a 2
85.m even 8 2 2312.1.f.b 2
85.p odd 16 8 2312.1.p.e 8
120.m even 2 1 1224.1.s.a 2
136.j odd 4 1 inner 3400.1.y.a 2
255.i odd 4 1 1224.1.s.a 2
340.n odd 4 1 544.1.n.a 2
680.k odd 2 1 2312.1.j.b 2
680.t even 4 1 3400.1.bc.b 2
680.bc odd 4 1 136.1.j.a 2
680.bc odd 4 1 2312.1.j.b 2
680.be even 4 1 544.1.n.a 2
680.bl even 4 1 3400.1.bc.a 2
680.bq odd 8 2 2312.1.e.a 2
680.bq odd 8 2 2312.1.f.b 2
680.co even 16 8 2312.1.p.e 8
2040.cg even 4 1 1224.1.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.j.a 2 5.b even 2 1
136.1.j.a 2 40.e odd 2 1
136.1.j.a 2 85.j even 4 1
136.1.j.a 2 680.bc odd 4 1
544.1.n.a 2 20.d odd 2 1
544.1.n.a 2 40.f even 2 1
544.1.n.a 2 340.n odd 4 1
544.1.n.a 2 680.be even 4 1
1224.1.s.a 2 15.d odd 2 1
1224.1.s.a 2 120.m even 2 1
1224.1.s.a 2 255.i odd 4 1
1224.1.s.a 2 2040.cg even 4 1
2312.1.e.a 2 85.m even 8 2
2312.1.e.a 2 680.bq odd 8 2
2312.1.f.b 2 85.m even 8 2
2312.1.f.b 2 680.bq odd 8 2
2312.1.j.b 2 85.c even 2 1
2312.1.j.b 2 85.j even 4 1
2312.1.j.b 2 680.k odd 2 1
2312.1.j.b 2 680.bc odd 4 1
2312.1.p.e 8 85.p odd 16 8
2312.1.p.e 8 680.co even 16 8
3400.1.y.a 2 1.a even 1 1 trivial
3400.1.y.a 2 8.d odd 2 1 CM
3400.1.y.a 2 17.c even 4 1 inner
3400.1.y.a 2 136.j odd 4 1 inner
3400.1.bc.a 2 5.c odd 4 1
3400.1.bc.a 2 40.k even 4 1
3400.1.bc.a 2 85.i odd 4 1
3400.1.bc.a 2 680.bl even 4 1
3400.1.bc.b 2 5.c odd 4 1
3400.1.bc.b 2 40.k even 4 1
3400.1.bc.b 2 85.f odd 4 1
3400.1.bc.b 2 680.t even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
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